5. DYNAMICS AND MELTING IN MODEL

HYDROGEN FLUORIDE CLUSTERS

5.1 INTRODUCTIO~

The behaviour of simple liquids when they freeze (or the behaviour of a solid as it

melts) has been the subject of considerable research. The interest in the study of this phase

transition is due to its overwhelming importance in a wide variety of physical situations

such as solidification [155], crystallization [156], nucleation [157], glass formation [73,74j,

etc.

The simplest situation arises when a liquid is cooled adiabatically [158]- that is to say

that at every stage it is in thermal equilibrium with the environment. One then usually

obtains a crystalline solid, and the phase transition itself is first-order, with a divergence

in specific heat [159]. Contrastingly, if the cooling rate is very rapid compared to the rate

of nucleation, then the same liquid can go into a metastable state which mayor may not

have crystalline order. A disordered metastable state of the solid is termed a glass [73],

and this transition is sometimes called the liquid ~ glass transition [73,74]. The nature

of this transition is poorly understood [160], largely because of the fact that unlike the

first-order liquid to crystal transition, the liquid to glass transformation is continuous as

far as volume, entropy and similar thermodynamic quantities are concerned [159]. The

viscosity, however, shows a sharp increase- typically from about 10-2 poise for the liquid,

to about 1013 poise for the glassy state. Quantities like the thermal expansion coefficient,

the heat capacity and the compressibility also show sharp changes near the glass transition

temperature.

Molecular dynamics simulation has been useful in studying glass formation [160,161J

51

and crystallizat.ion in simple as well as complex fluids. Hmvever, simple atomic liquids like

Ar are not transformed into a glassy state by laboratory experiment [160,162]. The exis­

tence of glassy state for such fluids has only been demonstrated by computer experimen.ts

in which the rate of quenching is roughly 4 to 6 orders of magnitude faster than any exper­

imental realized rate [161]. The glassy state produced by computer simulations for s~mple

liquids are not stable -these eventually go into the crystalline state [160]. It is widely

believed [162] that a glassy state in atomic systems like Ar is difficult to achieve because

of the isotropic nature of atomic interaction. On the other hand, experiments as well as

computer simulations show the existence of glass formation in binary fluid mixtures and

molecular systems [162]. The anisotropic nature of molecules give rise to strong coopera­

tive effects which is believed to be responsible for glass formation in the system. However,

owing to the large number of particles considered in the simulation of bulk liquids, the

time scale over which the computer simulations can be carried out is small compared to

the experimental time scale.

This has motivated us to study the solid ~ liquid transition in small dipolar hydrogen

fluoride(HF) clusters and to look at the possibility of glassy structures in finite systems.

HF is a strongly anisotropic molecule (perhaps the most polar covalent diatomic) and can

be taken as prototypical system for studying the effects of anisotropy and directionality in

clusters. (HF)n clusters would share some aspects of complex fluids because of associative

character of the hydrogen bonding. By focusing upon clusters with fewer molecules, we

are able to perform simulations over a relatively longer timescale.

Studies on molecular clusters as such are few compared to atomic systems. However,

it is relatively easier to produce molecular clusters experimentally, as compared to weakly

bonded rare gas clusters. Alkali halides have been a subject of much interest [lb3-167] in

the context of studying molecular clusters. Martin [165-167] and \Velch et. al. [163,164]

studied the geometries and vibrational properties of small and intermediate size alkali

52

halide clusters. Luo, Landamann and Jortner [168] studied dynamics in (NaCI)n and

shown that the behaviour depends strongly on the size of the clusters: small clusters seem

to undergo simple isomerization, whereas large clusters exhibit freezing/melting similar to

atomic systems. Rose and Berry [15] have studied (KC1)n clusters in the context of freezing,

melting and possibility of glassy structures. They have computed the configurational

density of states, and applied this to evaluate thermodynamic quantities. Wales and

Ohmine [17] have studied solid liquid transition in small water clusters. They mapped the

dynamics and properties in these clusters to the underlying potential energy surface and

have calculated various thermodynamic functions from the distribution of local minima

and found the appropriate features of melting in finite systems. Veigiri and Farantos [16]

have studied dynamics of small water clusters of sizes 3-8. They have investigated t.he

minimum energy structures and have found the tetramer and octamer clusters to be more

stable than their neighbours.

In this chapter, we study the detailed dynamics of (HF)n clusters for n = 7 -13, 15. In

addition to characterising clusters through the energetics, we also examine the dynamics

through power spectra and compute the largest Lyapunov exponent. Such studies have

not been yet done on molecular clusters. Our work address the following issues.

Are there any magic number HF clusters with unusual stability, analogus to magic

number rare-gas atomic systems?

What are the minimum energy configurations of HF clusters? What is the nature of

energy landscape of the potential energy surface of HF clusters?

Do finite HF clusters undergo a phase change as has been seen in finite atomic system?

Is the maximal Lyapunov exponent(MLE) an unambiguous characteristic of phase

change as one goes from hard potential (rare gas clusters) to more soft potential like in

53

HF clusters?

This chapter is organised as follows. In the next section we discuss modeling the

potential in molecular systems. The interaction potential function and details of the com­

putational methods are discussed in section 5.3. In section 5.4, we discuss results of the

simulations in terms of stability, caloric curves, root mean squared bond fluctuations, max-

imal Lyapunov exponent and power spectra. Conclusions are summarized in section 5.5.

5.2 MODELING POTENTIAL IN MOLECULAR SYSTEMS

Molecular systems are not rigid bodies in any sense [89]: they consist of fundamental

particles like electrons, protons and neutrons interacting via inter- and intra-molecular

forces. However, the forces acting within the molecules are of of higher order magnitude

than acting between the molecules. In other words, the amplitude of vibration is quite

small compared to molecular dimensions. So in practice, the intramolecular bonds are

kept fixed and the molecule is treated as a single rigid unit. The simplest way of modeling

[89] a molecule is to position each atom at a fixed bond length from every other atoms in

the molecule as shown in Fig. 5.1. The total potential is a sum of pairwise interactions

arising because of sites a of molecule i and b of molecule j. The interaction depends on

the relative positions of molecules Ti and G, and their relative orientations, which can be

specified therough (j and </J angles

V(Ti:rj,B,</J) = I:I:Vab(Tab) (5.1) a b

where Vab represents the pairwise interaction between sites a and b and Tab = ITi - Til. The interaction sites are usually centred on the positions of nuclei in the real molecule.

(This form of the model is known as site-site interaction.) The total interaction becomes

the sum of all pair-wise potentials. The short range site-site interaction can be expressed

54

0::7

0::2 -- ....

------"'7 ---- .... 1::>:::2 ' ----"-- ----- " , , -- , -- ..... -- ',--- ., -- ....

b=:1

Pig 5.1 Schematic /igure for modeling a diatomic moleCule.

j j j

j j j j j j j j j j j j j j j j j j j j

j j j j j

j j j

j

j j j

j j j j j

j j j

j j j

j

j j j j j j j j j j

j j j j j

j j j

j

j j j

in the Lennard-Jones form. The energy and length parameters can be suitably chosen

for interaction between identical atoms of a pair of molecules. The parameters for ,mlike

atoms in different molecules can be derived from Lorentz-Berthelet mixing rules [89]. For

example, in HF, the parameters are given by

1 aHF = i(aHH + aFF) (5.2)

fHF = (fHHfFF )1/2 (5.3)

The electrostatic interaction between polar molecules can be represented through partial

charges so as to reproduce the experimental dipole moments of the molecules in the gas

phase. The position of the charges and Lennard-Jones sites can be at the same place in

order to minimise computational effort.

5.3 POTENTIAL FUNCTION AND METHODS USED IN CLUSTER DYNAMICS

The HF molecule is treated as a rigid rotor. Each molecule is represented by two

potential sites namely the positions of hydrogen and fluorine. The intermolecular potential

for a pair of molecules i and j is thus of the form

(5.4)

where a and b are atoms on different molecules, rij is the distance between the atom a of

ith and b of jth molecule. qf, q~ represent the charges on atom a of molecule i and atom b

of molecule j respectively. The parameters qi, q~, fO, Cab, aab are given in Table 5.1. Thus

the potential is essentially a combination of long-range Coulombic terms and Lennard-

Jones form for short range interactions. The parameters Cab and aab are built up from H-H

and F -F interaction as described above. More accurate forms of the HF -HF intermolecular

55

H

F

Table 5.1: Potential parameters for model hydrogen fluoride.

qj in coulomb OJ III nm Cj / k~ in K ------------------------------

+0.2021 0.281 8.6

-0.2021 0.283 52.8

potential are known [108], but they do not have a computationally convenient form. The

present potential is very similar to the intermolecular sit.e-site potential of HF developed

by Brobjer and Murrell [169J. The essential features of a hydrogen-bonded fluid have been

incorporated; however, this potential has not been used so far to compute quantities to be

compared to experiments.

In order to integrate the equations of motions we follow the method for linear molecules

outlined in Ref. [89]. The molecular motions are divided into the translation of the centre

of mass and rotation about the centre of mass. Translational motions are handled similar

to the atomic case: the force acting on the centre of mass of the molecule h is given by

h = - L vV(ri;) = mir: i<;

where mi is the mass of a single molecule. The equations of motions are

(5.5)

(5.6)

(5.7)

Rotational motion is governed by the torque ii about the center of mass. The torque

acting on molecule i is given by ii = Ea d;.a x ha, ha being the force acting on molecule

i at atom a, J;.a being the position of atom a measured from the center of mass of the

molecule. The above equations can be simplified and written as

(5.8)

where ei is the unit vector along the molecular axis, g:l. = Ea dia X ha is the component

of g: along the axis perpendicular to the molecular axis. This gives, for the equations of

motion

56

(5.9)

(5.10)

where Ii is the moment of inertia of molecule i. The first term in the above equation

represent the force for the rotation of the molecule while .\ei term represents the force

which constrains the bond length. These equations are applied to each molecule.

The equations of motions are integrated using the leap-frog algorithm with. a time step

of 3.125 X 10-16 s. The total energy is conserved to within 0.01 %.

5.4 RESULTS AND DISCUSSIONS

A. Minimum energy structures

In this chapter we have studied (HF)n clusters of sizes n = 7-13, 15. The dynamics

of the clusters were studied by integrating equations of motions with the trajectories of

absolute minimum structures obtained as described below. The desired temperature can

be attained by increasing or decreasing the kinetic energy of all the particles. Sufficient

time is allowed at each temperature for equilibration (5 X lOs MD steps) before integra.ting

the equations of motions for 1.2 X 106 MD for computing ensemble averages.

In order to find the minimum energy configurations for the clusters studied, we adopt

the molecular dynamics procedure of Stillinger and Webber [37,64] as follows. Trajectories

are run with random initial structures having linear and angular momentum equal to

Zero. Then, by quenching the kinetic energy, several minima are approached. Quenching

was done very slowly and sufficient time was allowed in between quenches so that the

system was able to locate its global minimum. This process is repeated with different

initial conditions for several trials of heating and cooling for each cluster. By comparing

the energy of trajectories so obtained, we were able to assign the global minimum energy

configurations to each system. However, there is a likelihood that the real global minimum

57

structures may be missed by this technique. The different structures so obtained are

analysed by studying root mean squared distance, angular distribution etc.

While atomic clusters are well-known to have magic numbers [1] for stability (for

example Ar13, Arss and Ar147 are stable compared to their immediate neighbours like

Ar12, Ar14, ArS4), similar magic numbers are not known in many molecular systems. Magic

number effects in rare-gas clusters are largely geometric in origin-13, 55, 147 etc. being the

number of Mackay icosohedra [66]. For molecular clusters, it is not clear if such geometric

effects effects affect stability, although Rose and Berry [15] have indicated that there may

be magic number for KCI and other ionic halides. Shown in the Fig. 5.2 is the binding

energy per molecule for various clusters- the binding energy increases monotonically with

n. The increase in stability of the clusters with increase in cluster size can be interpreted

as arising from the increase in the number of hydrogen bonds in the system. Our present

result does not indicate any magic number for (HF)n.

Rose and Berry [15] located the minima of small KCI clusters using the same MD

method as well as the conjugate gradient technique [65]. They showed that small clusters

like KCh2 could find the way to the global minimum structures even with very high rate

of quenching. However, the answer to the question of whether small clusters can go to

a metastable state if quenched rapidly is still not clear, because of the lack of extensive

studies on such system. We are extending our present study to larger clusters like (HF)ss

in order to look for possible glassy structures.

B. Caloric Curve

In studying solid liquid phase change in 'clusters, it is usual to compute the caloric curve

which is a plot of total energy against the temperature of the system. In MD simulation

58

-11 I I I I

o -10.5 t- -

-10 t-

>. 0 01

'-I Ql -9.5 - 0 c Q)

01 C

. .-1

-"0 -9 c • .-1 0 .0

-8.5 t- -

-8 I- -

-7.5 I I I I

7 8 9 10 11 12 13 14 15 cluster size, N

Fig. 5.2 Binding energy per molecule of the absolute minimum vs. cluster size in dipolar HF

clusters.

the internal temperature of the system can be calculated from equipartition theorem. The

temperature, T of N molecule cluster is given by

T = 2Ekin (6N - 6)kb

(5.11)

where Ekin is the kinetic energy of the system averaged over the entire trajectory, kb = 1.381 X 10-23 erg/K is the Boltzmann constant. The caloric curve thus obtained shows

.. low energy and high energy regions and the onset of phase change is marked by change in

slope.

The caloric curve for all the clusters studied here are shown in Fig. 5.3. One can note

from this figure that the temperature, T of the system increases rapidly with internal

energy until the cluster starts melting, while this increase in temperature is slow in the

liquid phase. The melting temperature, Tm which is marked from the change in slope in

the caloric curve increases with cluster size. This can again be accounted for the increase

in the activation energy in the transition state due to increase in number of hydrogen

bonding in the system.

One more striking feature of the caloric curve is the absence of bimodal behaviour in

the kinetic energy distribution. Rare-gas clusters like Ar13 and Ar55 exhibit dynamical

coexistence where the clusters fluctuate between solid and liquid phase which in turn give

rise to bimodal behaviour in total kinetic energy distribution of the system. Berry and

coworkers [41] and later in a classic study, Labasttie and Whet ten [135] assigned this

behaviour to finite size effects and mapped this to the underlying potential landscape

of the system. In the present study we have probed different energy regions carefully,

particularly when the cluster starts melting, and have found no such bimodal behaviour.

59

120

100

80

60

40

20

o -11

+ +

n=7 0 n=8 + n=9 0

n=lO x n=l1 A

n=12 lie

n=13 0

n=15 +

+ +

+

+ +.

+

-10

+ +

+ .* +

X

X X

+0 *0 o

o

A X A/

o 000

o 0080 0

o o<f>

lie

)(

x x~

XX X t:. o

xx 80 )0( 0

x 0 x 0 x 0 x 0

+ o

o 0

+

-9 -8 -7 -6 energy/Kcal/mol/molecule

o

o

o

t:. t:.

o

A A lie t:.

t:. A

o

o o

-5

Fig. 5.3 Caloric curve for (HF)n clusters of sizes n = 7 - 13 and 15.

-4

C. Lindemann criterion for meIting(o)

The root mean squared fluctuation of the average intermolecular distance( 6) measures

structural changes in a more sensitive manner than the caloric curve. This is the Ljnde-

mann index and can be defined as

(5.12)

The 0 implies averaging over the entire run and rij is the distance between center of mass

molecules i and j, N is the number of molecules. Alternatively, ignoring the constraints of

the molecular bond lengths, 6 can be measured entirely from atomic coordinates as defined

in Eq. 2.17. We use both definitions and 6 obtained from either of these ways shows the

onset of melting at same temperature.

Shown in Fig. 5.4 the 6 using molecular coordinates versus total internal energy for

different clusters. Each curve shows three different regions. In the first region- the solid

state, 6 varies slowly with energy and as the cluster melts 6 varies rapidly, increasing above

0.1, which is the Lindemann criterion for melting in the system. The third region corre­

sponds to the liquid state where again 6 varies slowly with energy. It is to be noted here

that the plot of 6 verses the total energy is not very sharp near the melting temperature as

it happens in the study of rare-gas clusters. The intermediate energy region when clusters

start melting extends over a relatively wide range of energy and becomes a problematic

in defining precisely the. transition temperature Tm. However, the Lyapunov exponent

described below defines Tm more precisely.

60

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

o -11

n=7 <> n=8 + n=9 0

n=10 x n=11 l>.

n=12 lIE n=13 <> n=15 +

-10

+ + +

+

+

+

+ +

lIE

<>

S

<> X A

'+x

+ x 0

0 + ~ ....

00 <> 0 ...

<> 0 X

<>

00>

<> o04f>0

-9 -8 -7 -6 en~rgy/Kcal/mo~/molecule

o

-5

Fig. 5.4 Root mean squared fluctuation vs. total internal energy for all clusters studied here.

-4

D. Maximal Lyapunov Exponent(MLE)

We study the dynamics in (HF)n clusters and characterise the phase transition in the

system using MLE (definition of MLE is given by Eq. 3.9 in Chapter 3). For a N diatomic

~igid rotor molecular cluster, there 5N degrees of freedom owing to the constraint on bond

length. Thus the phase space of the system is ION dimensional. However, because of 7

conserved quantities (total energy, E, three linear and three angular momenta) there are

(5N -7) positive Lyapunov exponents. We calculate the largest of these defined by Eg,. 3.9.

Let £(t) represent a point in the phase space and :z7(t) be another close by point at a

distance ~x-(t). If the time derivative of £(t) is given by d1tt) = F(i, t) then

d~x-(t) = dx~t) _ dx(t) = F(:C'(t)) _ F(i(t)) dt dt dt (5.13)

Expanding F(x'(t)) upto first order one gets

d~x(t) = aft A -( )

dt ax ux t (5.14)

where ~~ is the Jacobian matrix.

Thus in order to follow the time evolution of ION points in the phase space one need

to solve the above equations.

Linearization of the equations of motion for the molecular cluster is somewhat more

complicated than the atomic case. In fact setting up the equations of motion and the Jaco­

bian during actual computation is a tricky matter. The ION phase space variables in this

system are 3N centre of mass coordinates and 3N conjugate momenta 2N bondvector co-

ordinates and the corresponding 2N conjugate momenta. However, in actual computation

we prefer to work in atomic coordinates in calculating forces or the Jacobian.

61

c·

The equations of Jacobian in terms of atomic coordinates can be written as follows. Let

~Xi, ~Yi, ~Zi are difference position variables and ~VXi, ~VYi, ~VZi are the corresponding

momenta. The expression for these variables are

(5.17)

(5.18)

These expressions are transformed into the molecular coordinates to follow the time evo-

lution of the phase space in the same manner as we use in solving equations of motions

62

for rigid diatomic molecules.

The MLE verses the total energy for clusters n = 7-12 are shown in Fig .S.5a and

for n = 13, 15 are shown in Fig. 5.5b. In contrast to 6, which varies smoothly (see

Fig. 5.4) with energy for HF clusters, ~he MLE shows discontinuity in slope at a well

defined temperature. Since this occurs at the temperature when 6 ~ 0.1) we prefer to

assign T m more unambiguously using MLE in the clusters.

In small clusters like n = 7-12, the variation of MLE with energy is continuous and the

plot has a knee near the transition temperature, Tm (see Fig. 5.5a). In larger clusters like

n=13, 15, this variation is very sharp and Tm can be defined precisely. Our earlier work

on atomic clusters (see Chapter III) has shown similar correspondence with the phase

transition. Although here we have not computed total potential energy distribution, we

expect that as in the atomic case, the phase change is accompanied by the change in phase

space available to the system.

The potential function used for HF clusters is quite different compared to the interac­

tion in rare-gas clusters -the former has electrostatic part in addition to Lennard-Jones

form which makes the potential more soft. Also here the range of potential is quite large

compared to rare-gas system. However, the MLE still define" the phase transition in molec­

ular cluster in analogy with atomic system and hence can be taken as an unambiguous

characteristic of phase transition.

E. Angular distribution

The distribution of angles between the nearest neighbours gives further indication to

the phase change in the clusters. In the crystalline phase the molecules in the clusters are

63

+-' C Q)

C o 0. X Q)

:> o c

,::1 0. IU :>, H

1.2

1

0.8

0.6

0.4

0.2

o -11

n=7 0 n=8 + n=9 0

n=10 x n=l1 D.

n=12 )I(

lIE

lIE lIE lIE lIE ~

f -10

>t x x

x

fA

fA

xx x o

fA

'S<

o

fA

XX~ x

x x

0 x 00

0

o o

o +

X

+

-9 -8 -7 -6 en~rgy/Kcal/mol/molecule

-l> D.

D.

0

o

)I(

o

-5

Fig. 5.5a Plot of maximal Lyapunov exponent vs. total internal energy for clusters of size

n = 7 - 12.

-4

1.2 I I

0 n=13 0 0 0

n=15 + • o 0 <> + o 0 0 000

0 -00 0

1 r 000 0

0 -++

+ 0

+ 00 0

~ 0.8 -c: Q) + c: 0 + 0. + + x Q)

0.6 0 :> + 0 c:

-:J ++

0. 0 III ++ >. H 0.4 -+ 0

+ 0

+ 0.2 r + 0

0 0

0 Ir··

t i -.l -.l .1 I

-11 -10 -9 -8 -7 -6 -5 -4 energy/Kcal/mol/molecule

Fig. 5.5b Plot of maximal Lyapunov exponent vs. total internal energy for clusters of size

n = 13,15.

arranged in a definite order. The angles between the nearest neighbours are constrained.

However, in the liquid phase the intermolecular distances are large and the molecules are

arranged randomly with no particular choice between the angles between the neighbouring

molecules. Thus in the solid phase the distributions confines to certain particular angles

where as in the liquid phase the distribution shows possibility of every angles: this is

possible owing to the large distances between molecules in the liquid state. The angular

distribution for HF13 clusters for solid and liquid pha.se our shown in Fig. 5.6a and 5.6b

respectively.

F. Power Spectra

In analogue with atomic cluster, the power spectra of potential energy of a tagged I

molecule is defined as

S(f) = lim I.!. r dt~(t)exp( -ift)12 = Re[ roo dtC/c(t)exp( -ift)]. r-+oo T 10 10 (5.21)

where the subscript k refers to the kth molecule of the cluster, and V/c(t) = Lj# V(r/cj(t))

is the potential energy of the tagged molecule, k. The power spectra indicates temporal

correlation present in the system.

In atomic cluster we have observed the power spectra of a tagged atom to have 1/ f dependence over a wide frequency range (see Chapter II). The origin of such spectra is

explored by studying dynamics of mixed atomic clusters and this behaviour is seen to arise

because of the surface core motion.

The idea of exploring the intrinsic long range temporal correlation in the system

through tagged particle spectroscopy is due to Sasai, Ohmine and Ramaswamy [110]. In

their work on liquid water dynamics, they observed the power spectra of water molecule in

64

1 I IT I .- T I I I

0.8 -r--

c 0

• .-1 r--.., f-:J :0

0.6 . .-1 r--r--

~ .., r--III r--

• .-1 >--"0

>. .., r--rl . .-1 0.4 .0 IU .0

f-~

I- .--f--f-

0 .--~ 0. I-- f-.---

0.2 I--

~

- r--f--

>--

>-- I--

o I I. 1-+ rh J J

o 20 40 60 80 100 120 140 160 180 200 angle in theta

Fig. 5.6a Angular distributions for HF13 in the solid phase.

1 I T I I I I I T r-

~

f-- f--

,.....--,

0.8 -I

r- ,-c: 0 r-

-..-i ~ - f-- -,-

-::l .n I---..-i 0.6 r-H ~ r-til

-..-i -0 r- r-

>. I--

~ >--M -..-i 0.4 .n

t-~

r-10 .n r-0 r-H Q,

f--

0.2 ~ -r-

r--I--

o II I t .1. 1 rL o 20 40 60 80 100 120 140 160 180 200

angle in theta

Fig. 5.6b Angular distributions for HF13 in the liquid phase.

the liquid state to have 1/ f behaviour while corresponding spectra in simple atomic liquid

is of white noise type. It is believed that the energy relaxation processes in liquid water

is of non-Debye type and the different time scales arises because of many instantaneous

network structures due to hydrogen bonding in the system.

Thus one would expect to see such behaviour in hydrogen bonding cluster like (HF)n.

However, contrary to the expectation, the power spectra in small HF cluster has 1/ f dependence over a smaller frequency range compared to similar spectra i~ atomic clusters.

Shown in Fig. 5.7 is the power spectrum of tagged molecule in (HF)13 at T = 90Kin liquid

state. The spectrum has 1/ f dependence. over one decade of frequency and below an onset

frequency, fc( here ) the spectrum is flat. This is a striking feature as one would expect

the power spectra to have 1/ f character over wide range of frequency because of both

hydrogen bonding and surface core motion in these systems.

The above observation can be interpreted as follows. Unlike in liquid water where the

different time scales arise due to large number of networks, in small HF clusters this is

not the case. Because of small number of molecules present, very few network structures

are formed in small clusters. On the other hand, due to hydrogen bonding the motions in

such systems are less diffusional and as a result .the surface core motion is suppressed to a

large extent. Thus the motions are strongly correlated for short time scale while long-time

motions are not. We are extending our studies for larger clusters in order to see variation

of onset frequency (fc) on size ofthe cluster. In larger clusters, it is expected the spectrum

would have 1/ f behaviour over a wide range because of formation of large numbers of

network structures.

65

le+08

le+07

1e+06

o

100000 ~ ______ ~ ____ ~ __ ~~~~~-L~ ______ ~ ____ ~ __ ~ __ ~~~~~ ____ ~

0.01 0.1 frequency, f

1

Fig. 5.7 Power Spectrum of a tagged molecule in liquid phase of a HF 13 cluster at T = 90K.

5.5 CONCLUSION

In this chapter, we have studied the dynamics of small hydrogen fluoride clusters

when they undergo solid ~ liquid transition. The stability of these clusters in terms

of minimum energy configuration was investigated using MD method. Our results show

that the stability of (HF)n increases with cluster size-,- which is in complete contrast to

rare gas clusters where some particular clusters are relatively stable than their immediate

neighbours. However, it is not clear whether such geometric effects are present in molecular

clusters, like (HF)n.

We have computed the maximal Lyapunov exponent (MLE) and found MLE defines the

phase transition more precisely than Lindemann index. The power spectrum of individual

potential energy fluctuations in (HF)n clusters is seen to have 1/ f dependence over a

relatively smaller range of frequency compared to that in atomic clusters.

66